in any event vs at any rate

in any event and at any rate both are adverbs.

Word	Noun	Adjective	Verb	Adverb
in any event	No	No	No	Yes
at any rate	No	No	No	Yes

As adverbs, in any event and at any rate are synonyms defined as:

in any event and at any rate: used to indicate that a statement explains or supports a previous statement

Other synonyms of in any event include anyhow, anyway, anyways, in any case.

in any event (adverb)	at any rate (adverb)
used to indicate that a statement explains or supports a previous statement	used to indicate that a statement explains or supports a previous statement
	if nothing else (`leastwise' is informal and `leastways' is colloquial)